Binomial-law conjecture for class and norm factors in cyclic p-extensions

Let $L/K$ be any cyclic $p$-extension of degree $p^N$, with $N \geq 1$, and Galois group $G$. Assume that $L/K$ is ramified at $r \geq 1$ places of $K$, totally ramified in $L/K$. Binomial-law conjecture. The orders of each of the two factors, class and norm, in the $i$-sequence

$$\#({\mathcal H}_L^{i+1}/{\mathcal H}_L^i)$$

follow binomial laws as $i$ increases, based on the following probabilities: for $c \in {\mathcal H}_K$, the probability that the $p$-class of ${\bf N}_{L/K}({\mathfrak C})$ equals $c$, for an ideal ${\mathfrak C}$ of $L$, is

$$\frac{1}{\# {\mathcal H}_K};$$

and, for $\gamma \in G^{r-1}$, the probability that $\omega_{L/K}(x)=\gamma$, for $x \in K^\times$ that is a local norm at all non-ramified places, is

$$\frac{1}{p^{N(r-1)}}.$$

The conjecture proposes a probabilistic model for the successive class-group filtration factors and norm factors; the source gives heuristic motivation but no proof or resolution.